Skip to main content
eScholarship
Open Access Publications from the University of California

Combinatorial Theory

Combinatorial Theory banner

Numerical semigroups, polyhedra, and posets I: the group cone

  • Author(s): Kaplan, Nathan;
  • O'Neill, Christopher
  • et al.

Published Web Location

https://doi.org/10.5070/C61055385Creative Commons 'BY' version 4.0 license
Abstract

Several recent papers have explored families of rational polyhedra whose integer points are in bijection with certain families of numerical semigroups. One such family, first introduced by Kunz, has integer points in bijection with numerical semigroups of fixed multiplicity, and another, introduced by Hellus and Waldi, has integer points corresponding to oversemigroups of numerical semigroups with two generators. In this paper, we provide a combinatorial framework from which to study both families of polyhedra. We introduce a new family of polyhedra called group cones, each constructed from some finite abelian group, from which both of the aforementioned families of polyhedra are directly determined but that are more natural to study from a standpoint of polyhedral geometry. We prove that the faces of group cones are naturally indexed by a family of finite posets, and illustrate how this combinatorial data relates to semigroups living in the corresponding faces of the other two families of polyhedra.

Keywords: Polyhedron, numerical semigroup.

Mathematics Subject Classifications: 52B05, 20M14

Main Content
For improved accessibility of PDF content, download the file to your device.
Current View